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Step-by-step explanation:
[tex] \large{ \sf \underline{Solution - }}[/tex]
To rationalise :
[tex] \sf \implies\cfrac{3 + 2 \sqrt{5} }{5 - 2 \sqrt{3} } [/tex]
➢ Here the denominator is in the form of (a-b). Rationalising factor of (a-b) is (a+b). So, rationalising factor of (5-2√3) is (5+2√3). We will multiply (5+2√3) with both the numerator and denominator to rationalise the denominator.
Now,
[tex] \sf \implies\cfrac{3 + 2 \sqrt{5} }{5 - 2 \sqrt{3} } \times\cfrac{5+2 \sqrt{3} }{5+2 \sqrt{3} } [/tex]
Combine fractions
[tex]{ \sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{(5 - 2 \sqrt{3})({5+2 \sqrt{3}) } }}[/tex]
We know that,
[tex]\sf \implies (a + b)(a - b) = a^{2} - b ^{2} [/tex]
So,
[tex] \sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{(5) ^{2} - (2 \sqrt{3})^{2} }[/tex]
Simplify the denominator,
[tex] \sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{25 - 12 }[/tex]
[tex] \sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{13 }[/tex]
Now, simplify the numerator,
[tex] \sf \implies\cfrac{3(5 + 2 \sqrt{3}) + 2 \sqrt{5} (5 + 2 \sqrt{3} ) }{13 }[/tex]
[tex] \sf \implies\cfrac{15 + 6 \sqrt{3} + 10 \sqrt{5} + 4 \sqrt{15} }{13 }[/tex]
Hence,
On rationalising we got,
[tex] \bf \implies\cfrac{15 + 6 \sqrt{3} + 10 \sqrt{5} +4\sqrt{15} }{13 }[/tex]